Nonlinear Joint Spectral Radius

Abstract: We introduce a nonlinear extension of the joint spectral radius (JSR) for switched discrete-time dynamical systems governed by sub-homogeneous and order-preserving maps acting on cones. We show that this nonlinear JSR characterizes both the asymptotic stability of the system and the divergence or convergence rate of trajectories originating from different points within the cone. Our analysis establishes upper and lower bounds on the nonlinear JSR of a sub-homogeneous family via the JSRs of two associated homogeneous families obtained through asymptotic scaling. In the homogeneous case, we develop a dual formulation of the JSR and investigate the equality between the joint spectral radius and the generalized joint spectral radius, extending classical results from linear theory to the nonlinear setting. We also propose a polytopal-type algorithm to approximate the nonlinear JSR and provide conditions ensuring its finite-time convergence. The proposed framework is motivated by applications such as the analysis of deep neural networks, which can be modeled as switched systems with structured nonlinear layers. Our results offer new theoretical tools for studying the stability, robustness, and convergence behavior of such models.